The basic reproduction number in epidemic models with periodic demographics

The epidemic growth model is an important tools used in predicting the future of a population and the spread of disease in the population. An epidemic model is usually formed in a differential equation or a system consisting several differential equations. The biological complexity in the underlying population affects the complexity of the epidemic model. One example of biological complexity is the Allee effect which reflects the critical density dependent of the population growth. In this paper we discuss a Logistic epidemic by considering this Allee effect on the population. Dynamic analysis is performed by determining fixed point and its stability analysis in crisp condition. We found the Basic Reproduction Ratio (BRR) for the model. The properties of the solution of the model are explored by the use of its numerical solution. Since we also consider the fuzziness of parameters and variables in the model, the numerical solution is generated using a modified Runge-Kutta method. This is done to explore the effect of inaccuracy and uncertainty which often occur in epidemiological problems.

Comparison of deterministic and stochastic SIS and SIR models in discrete time

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infected-susceptible) epidemic model with n patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. The basic reproduction number ${\cal R}_0$ for the model is determined. It is then shown that when the disease-free equilibrium is stable (${\cal R}_0 < 1$) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (${\cal R}_0 > 1$) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (${\cal R}_0 < 1$) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (${\cal R}_0 > 1$). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.

Mathematical perspective of Covid-19 pandemic: Disease extinction criteria in deterministic and stochastic models.

Stability analysis of an HIV/AIDS epidemic model with treatment

&lt;abstract&gt;&lt;p&gt;Pandemics, such as Covid-19 and AIDS, tend to be highly contagious and have the characteristics of global spread and existence of multiple virus strains. To analyze the competition among different strains, a high dimensional SIR model studying multiple strains' competition in patchy environments is introduced in this work. By introducing the basic reproductive number of different strains, we found global stability conditions of disease-free equilibrium and persistence conditions of the model. The competition exclusion conditions of that model are also given. This work gives some insights into the properties of the multiple strain patchy model and all of the analysis methods used in this work could be used in other related high dimension systems.&lt;/p&gt;&lt;/abstract&gt;

This paper develops a fractional-order epidemic model (SVEIR) incorporating a generalized incidence rate, analyzed within the framework of the Caputo fractional derivative. The foundational properties of the model, including existence, uniqueness, non-negativity, and boundedness of solutions, are rigorously established to ensure its well-posedness. The basic reproduction number $R_0$ is computed using the next-generation matrix method, providing threshold criteria for disease elimination and persistence. The model exhibits two equilibria: the disease-free equilibrium and the endemic equilibrium. Through stability analysis, we prove the global stability of these equilibria by constructing suitable Lyapunov functions and applying LaSalle's invariance principle. To validate the theoretical results and examine the effects of various epidemiological parameters, numerical simulations are performed using MATLAB. These simulations provide deeper insights into the dynamic behavior of the proposed fractional model, highlighting its potential applicability in understanding and controlling disease spread.

A co-infection model between HIV and COVID-19 that takes into account COVID-19 vaccination and public awareness is discussed in this article. Rigorous analysis of the model is conducted to establish the existence and local stability conditions of the single-infection models. We discover that when the corresponding reproduction number for COVID-19 and HIV exceeds one, the disease continues to exist in both single-infection models. Furthermore, HIV will always be eradicated if its reproduction number is less than one. Nevertheless, this does not apply to the single-infection COVID-19 model. Even when the fundamental reproduction number is less than one, an endemic equilibrium point may exist due to the potential for a backward bifurcation phenomenon. Consequently, in the single-infection COVID-19 model, bistability between the endemic and disease-free equilibrium may arise when the basic reproduction number is less than one. From the co-infection model, we find that the reproduction number of the co-infection model is the maximum value between the reproduction number of HIV and COVID-19. Our numerical continuation experiments on the co-infection model reveal a threshold indicating that both HIV and COVID-19 may coexist within the population. The disease-free equilibrium for both HIV and COVID-19 is stable only if the reproduction numbers are less than one. Additionally, our two-parameter continuation analysis of the bifurcation diagram shows that the condition where both reproduction numbers equal one serves as an organizing center for the dynamic behavior of the co-infection model. An extended version of our model incorporates four different interventions: face mask usage, vaccination, and public awareness for COVID-19, as well as condom use for HIV, formulated as an optimal control problem. The Pontryagin's Maximum Principle is employed to characterize the optimal control problem, which is solved using a forward-backward iterative method. Numerical investigations of the optimal control model highlight the critical role of a well-designed combination of interventions to achieve optimal reductions in the spread of both HIV and COVID-19.

This chapter examines mathematical models that describe disease spread, beginning with the classical Kermack–McKendrick SIR model, which divides populations into Susceptible, Infected, and Removed compartments. It explains how the basic reproduction number (R0) determines whether an epidemic grows or dies out, and explores strategies to reduce R0, such as vaccination and social distancing. The chapter extends to SIR model variations, including the SEIR and SIS models, which account for incubation periods and diseases without lasting immunity, respectively. It also analyses real-world applications, such as Covid-19 lockdown measures and foot-and-mouth disease control, where mathematical models were used to guide policy decisions. It introduces structured models, where parameters may vary depending on the age of an individual or on their behaviour. Understanding how drugs are absorbed in the body and how they tackle disease is the subject of the area of Pharmacokinetic/Pharmacodynamic (PK/PD) modelling and the key building blocks of this approach are presented. The chapter also introduces within-host models, which aim to capture how a virus grows in the host. Coupling such models with those at the population level can then capture key aspects of the multiscale complexity of disease spread.

A stochastic SIS epidemic model with saturating contact rate

We discuss the epidemic network model with infectious force in latent and infected period. We obtain the basic reproduction number and analyze the globally dynamic behaviors of the disease‐free equilibrium when the basic reproduction number is less than one. The effects of various immunization schemes are studied. Finally, the final sizes relation is derived for the network epidemic model. The derivation depends on an explicit formula for the basic reproduction number of network of disease transmission models.

Objective To develop a prognostic SIR model of the COVID-19 pandemic in Colombia. Materials and Methods A SIR model with a deterministic approach was used to forecast the development of the COVID-19 pandemic in Colombia. The states considered were susceptible (S), infectious (i) and recovered or deceased (R). Population data were obtained from the National Administrative Department of Statistics (DANE)—Population Projections 2018-2020, released in January 2020—, and data on daily confirmed cases of COVID-19 from the National Institute of Health. Different models were proposed varying the basic reproduction number (R0). Results Based on the cases reported by the Ministry of Health, 4 simulated environments were created in an epidemiological SIR model. The time series was extended until May 30, the probable date when 99% of the population will be infected. R0=2 is the basic reproduction number and the closest approximation to the behavior of the pandemic during the first 15 days since the first case report; the worst scenario would occur in the first week of April with R0=3. Conclusions Further mitigation and suppression measures are necessary in the containment and sustained transmission phases, such as increased diagnostic capacity through testing and disinfection of populated areas and homes in isolation.

We propose a stage-structured model of childhood infectious disease transmission dynamics, with the population demographics dynamics governed by a certain family and population planning strategy giving rise to nonlinear feedback delayed effects on the reproduction ageing and rate. We first describe the long-term aging-profile of the population by describing the pattern and stability of equilibrium of the demographic model. We also investigate the disease transmission dynamics, using the epidemic model when the population reaches the positive equilibrium (limiting equation). We establish conditions for the existence, uniqueness and global stability of the disease endemic equilibrium. We then prove the global stability of the endemic equilibrium for the original epidemic model with varying population demographics. The global stability of the endemic equilibrium allows us to examine the effects of reproduction ageing and rate, under different family planning strategies, on the childhood infectious disease transmission dynamics. We also examine demographic distribution, diseases reproductive number, infant disease rate and age distribution of disease, and as such, the work can be potentially used to inform targeted age group for optimal vaccine booster programs.

Due to the negative impact of infectious diseases on population growth, it is important to understand the dynamic behavior of such diseases. Mathematical deterministic SIR models are widely used to study the spread of infectious diseases. In real life, there is a lot of randomness and stochasticity such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. in the considered models. In that case, we must introduce Poisson noises into the population models for describing such discontinuous systems. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous SIR model disturbed by white noise, centered and non-centered Poisson noises. In the deterministic case there is a threshold of the system for an epidemic to occur, so called the basic reproduction number. Depending on the value of the reproduction number there is the disease-free equilibrium, or there is an endemic equilibrium, which implies the disease always remains. In the case of the autonomous stochastic SIR model, we study the asymptotic behavior of the solution to the corresponding system of stochastic differential equations around these points of equilibrium.

In this paper, a new epidemic SEIRS model with time delay on complex networks is proposed. Based on the mean field theory, the basic reproductive number and equilibriums of the model are derived. Moreover, the impact of the network topology and time delay on the basic reproductive number is analyzed. Theoretical analyses indicate that the basic reproductive number is dependent on the topology of the underlying networks. The time delay cannot change the basic reproductive number, but it can reduce the endemic level and weaken the epidemic spreading. The global asymptotically stability of the disease-free equilibrium and the permanence of epidemic are proved in detail. Numerical simulations confirm the analytical results.